Nilpotent Numbers Jonathan Pakianathan and Krishnan Shankar
نویسندگان
چکیده
Introduction. One of the first things we learn in abstract algebra is the notion of a cyclic group. For every positive integer n, we have Zn, the group of integers modulo n. When n is prime, a simple application of Lagrange’s theorem yields that this is the only group of order n. We may ask ourselves: what other positive integers have this property? In this spirit we call a positive integer n a cyclic number if every group of order n is cyclic. We define abelian and nilpotent numbers analogously. Recall that a group is nilpotent if and only if it is the (internal) direct product of its Sylow subgroups; see [7, p. 126]. This is not a new problem; the cyclic case is attributed to Burnside and has appeared in numerous articles, [9], [4], [1], [2]. The abelian case appears as a problem in an old edition of Robinson’s book in group theory; see also [6] and the nilpotent case was also done quite some time ago (see [5], [8]). In this article we give an arithmetic characterization of the cyclic, abelian, and nilpotent numbers from a single perspective. Throughout this paper Zn will denotes the cyclic group of order n.
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Nilpotent Numbers
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